Free energy of the concerted-exchange mechanism for self-diffusion in silicon

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DOI: 10.1103/PhysRevB.53.1310

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Free energy of the concerted-exchange mechanism for self-diffusion in silicon

Instituto de Fı´sica Gleb Wataghin, Universidade Estadual de Campinas, Unicamp, 13083-970 Campinas, Sa˜o Paulo, Brazil

S. Ismail-Beigi*and Efthimios Kaxiras

Department of Physics and Division of Applied Sciences, Harvard University, Cambridge, Massachusetts 02138

K. C. Pandey

IBM Research Division, T. J. Watson Research Center, Yorktown Heights, New York 10598

~Received 8 August 1995 !

The free energy of the concerted exchange mechanism for self-diffusion in silicon is estimated using the thermodynamic integration method and Monte Carlo~MC! simulations with an interatomic potential fitted to reproduce local-density-approximation calculations. Anharmonicity and relaxation are fully taken into account in the calculations, since the phase space is extensively explored by the MC simulations. The results indicate that the concerted exchange mechanism can have a significant contribution to the self-diffusion constant in silicon.

I. INTRODUCTION

Atomic diffusion in solids plays a very important role for various physical properties. In semiconductors, atomic diffu-sion is crucial for controlling doping, interfacial mixing, etc., which affect their electronic and optical properties. Silicon is by far the most extensively studied semiconductor. Neverthe-less, our understanding of self-diffusion in silicon at the mi-croscopic level remains an elusive issue. Since silicon is a covalent solid, it was expected that self-diffusion could only be mediated by native defects, such as vacancies and inter-stitials. Early theoretical studies1 found an activation en-thalpy for defect mechanisms to be in the range 3.5– 4.5 eV, in reasonable agreement with experimental data, which lie between 4.1 and 5.1 eV.2 Pandey has proposed a different mechanism, the concerted exchange ~CE!, in which, two neighboring atoms exchange positions along a specific path, without the intervention of a native defect.3The two atoms that are exchanging positions in the CE mechanism execute a complex motion in three-dimensional space in order to avoid large energy barriers. Along this path, bonds between the exchanging atoms and their nearest neighbors are succes-sively broken and reformed. Figure 1 shows the exchanging atoms in their equilibrium positions and at the saddle-point configuration. The calculated activation enthalpy of 4.3 eV for this mechanism3is similar to those of the defect mecha-nisms and lies in the range of the experimental data.

From the above discussion it is evident that, based solely on the enthalpy, it is not possible to determine the existence of a dominant microscopic mechanism responsible for self-diffusion in silicon. The experimental data show that the en-tropy contribution to the activation free energy for self-diffusion is substantial, approximately 7kB to 9kB.2 Thus,

since these microscopic mechanisms are very different in nature, theoretical estimates of the free energy and entropy could shed some light on the issue of the existence of a dominant self-diffusion mechanism. Quantities such as

inter-nal energy and enthalpy are functions of the positions and velocities of the atoms, and can be easily determined from the calculations. The free energy and the entropy, on the other hand, are much more difficult to determine, because they depend on the total phase-space volume accessible to the system.4 Estimates of the activation entropy for the de-fect mechanisms were;1kB.5Pandey and Kaxiras,6 apply-ing classical transition-rate theory7 to the ideal energy sur-face ~without relaxation! of the CE mechanism, obtained a lower bound for the entropy of this process of 3.3k_B. Sub-sequently, the same authors8 included full relaxation of the energy surface, as well as contributions of vibrations within the harmonic approximation, yielding an activation entropy of 6.3kB, which accounts for a substantial part of the

experi-mental data. More recently, Blo¨chl et al.,9 using parameter-free molecular-dynamics simulations, studied the defect me-diated mechanisms for self-diffusion and found that the interstitialcy mechanism yields a diffusion coefficient very close to experiment, whereas the vacancy mechanism yields a diffusion coefficient 3 orders of magnitude smaller.

Since in the previous estimates of the activation entropy of the CE mechanism the entire phase-space available for the system was not completely explored, it would seem appro-priate to readdress the question using a different methodol-ogy. Here we provide an accurate determination of the acti-vation free energy for the CE mechanism using the thermodynamic integration ~TI! method4,10–14 and Monte Carlo~MC! simulations to generate averages in the canonical ensemble. The TI method, initially proposed by Kirkwood,10 has been widely used over the years for a variety of problems such as vacancies in rare-gas solids,11 phase transitions,4,12 diffusion of impurities in Si,13 extended defects in metallic alloys,14etc. In the following we describe the method and the approximations used in the calculations.

II. SIMULATION METHOD

In our calculations, the interaction between Si atoms was modeled using an effective two- and three-body potential,15

53

which is based on a large database obtained from local den-sity approximation~LDA! calculations. This potential repro-duces the equilibrium lattice constant, the bulk modulus, and the cohesive energy of bulk Si, as well as the LDA results for the energy along two paths in the CE energy landscape: the migration path and a path perpendicular to the migration path, passing through the saddle point. These two paths rep-resent the most important features of the total-energy land-scape. The migration path is obviously important because it contains the saddle point configuration. The path perpendicu-lar to it is also important, because its curvature at the saddle-point is related to the activation entropy within the classical transition-rate theory.7Tests of this potential revealed that it also provides an excellent description of the fully relaxed saddle-point configuration, without having fitted it. Thus, it should be reliable for the type of calculations reported here. The constant volume Monte Carlo simulations were per-formed using a supercell geometry and periodic boundary conditions to simulate the infinite crystal. The Monte Carlo method was implemented according to the Metropolis algo-rithm. Typically, 104 MC steps were used to achieve equili-bration and averages were taken from samples with 53104 MC steps. The Helmholtz free energy difference between two states with energy E₀ and E₀1E₁ is given in the TI method by DF5

E

K

L

where E_l5E01lE1 and

^&

canonical ensemble:

^

&

E

where *@dX# represents an integral over the entire phase space available for the system. The energy E_ldoes not nec-essarily represent a physical system for 0,l,1. In most cases, the kinetic energy remains the same asl varies, since the overall number of particles in the system remains the same. Therefore, the kinetic energy does not contribute to the change in the free energy. Since this is the case here, we will not consider the kinetic energy in our calculations.

In order to calculate the activation free energy of the CE mechanism we considered the following potential energy, for a supercell of N atoms:

E_l5E₀1l«₁1~12l!«₂, ~3! where E0includes the potential energy of all the atoms in the

cell, except for those that are exchanging sites ~a total of

N22 atoms!, «₁ is the potential energy of the two exchang-ing atoms constrained to vibrate around the saddle-point con-figuration, and«2is the potential energy of the two

exchang-ing atoms vibratexchang-ing at the perfect crystal sites. Durexchang-ing the simulation, care must be taken to keep the two exchanging atoms on the saddle-point surface,8since otherwise the atoms would depart from the saddle-point configuration and move toward the perfect crystal geometry. In terms of the angular coordinates defined by Pandey to describe the CE mechanism,3 the following restriction is necessary near the saddle-point configuration: u52f12p/3. This constrains one degree of freedom in the system. To achieve the desired effect of keeping the two atoms on the saddle-point surface, an additional degree of freedom must be constrained; to this end, it suffices to constrain the center of mass of the two exchanging atoms to lie on the saddle-point surface, which near the saddle-point configuration coincides with the plane defined by their four neighbors @see Fig. 1~b!#.

It should be pointed out that the two pairs of exchanging atoms at the perfect crystal configuration and at the saddle-point configuration do not interact with each other, but they do interact with all the other atoms in the cell. In other words, forl50 one has N atoms oscillating around the per-fect crystal positions plus two atoms constrained at the saddle-point surface but decoupled from the rest of the sys-tem. Forl51, there are N22 atoms atoms vibrating around the perfect crystal positions interacting with two atoms con-strained at the saddle-point plane, plus two atoms oscillating around the ideal crystal positions but decoupled from the rest of the system. The atoms that are decoupled from the rest of the system are confined to a sphere, in order to prevent them from wandering freely throughout the system. This can be easily implemented in the MC simulations by not accepting moves that take the atoms out of the spheres. The spheres, whose radius is taken to be approximately the amplitude of the atomic vibrations at each temperature, are centered at the positions of the atoms for the perfect crystal and the saddle-point configurations. The integral in Eq.~1! is evaluated nu-merically using the Gauss-Legendre ~GL! quadrature method. Initially, a 54-atom supercell and a 10-point GL

FIG. 1. Representation of the neighborhood of the CE mecha-nism in ~a! the perfect crystal and ~b! the saddle-point configura-tions. The atoms exchanging sites are shown shaded. In ~a! the arrows indicate the motion of the atoms from the equilibrium to-ward the saddle-point point configuration. In ~b! the dashed lines indicate the plane where the two atoms at the saddle-point are re-stricted.

quadrature were used. The integrand in Eq.~1! turned out to be a very smooth function ofl. This test indicated that a GL quadrature with fewer points might be used without intro-ducing substantial numerical error. By reintro-ducing the number of points in the GL quadrature we were able to increase significantly the size of the supercell. The results presented here were obtained using a 128-atom supercell and a 5-point GL quadrature.

III. RESULTS

Figure 2 shows the evolution of the components of the potential energy «1 and «2, as well as the total potential

energy E_lduring the MC simulation for three values ofl at

a temperature of 1500 K. This figure contains simulation results for the smallest and largest values ofl in the 5-point GL quadrature, as well as for the middle-point value. It is important to note that even for the smallest and largest l values, the extreme cases when there is a pair of particles almost decoupled from the rest of the system, the simulations are quite stable. This is the reason why the thermodynamic average in Eq.~1!, which involves an integral of the quantity

]E_l

]l 5«12«2 ~4!

is a smooth function of l. As expected, however, the fluc-tuations are more pronounced for the smallest and largestl

FIG. 2. Evolution of the potential energy components «1, «2, and the total potential energy El ~see text! for ~a! the lowest (l 5

0.046 910 078!, ~b! the middle (l 5 0.5!, ~c! the largest (l 5 0.953 089 923!, values of the parameter l of a 5-point GL quadrature, at a temperature T51500 K. Energies are shown at intervals of 200 MC steps. The energy of the 128-atom supercell at 0 K is taken as the zero of the potential energy.

values. It should be noted that although the derivative in~4! does not depend explicitly onl, the canonical average of it does depend onl, since the atomic configurations during the simulations are generated using the potential energy E_l@see Eq. ~2!#. Another interesting feature is that while l varies from 0 to 1, the average of «1 remains approximately the

same. On the other hand, the average of «2 increases with l, that is the energy of the pair of atoms vibrating at the

perfect crystal sites increases as they are decoupled from the rest of the system.

We can now compare our results with the experimental self-diffusion data. At high temperatures, the diffusion coef-ficient can be written as16

D5D0e2DH/kBT, ~5!

where D0 is the pre-exponential factor and DH is the

vation enthalpy. In the case of the CE mechanism, the acti-vation enthalpy is just the difference in enthalpy between the saddle point and the perfect crystal configurations. The pre-exponential factor is given by

D05

1 6f za

2_neDS/k_{B, ~6!}

The factor f is the correlation factor between consecutive jumps, which is equal to 1 for self-diffusion ocurring through exchange events.17The coordination number z in the case of a diamond lattice is 4. The jump distance, a, is the inter-atomic distance in silicon. The factor n5

A

the attempt frequency for the CE mechanism, where m5mSi/2. The activation entropyDS is given by the sum of

two terms,

DS5DSconf1DSvib, ~7!

whereDSconf5kBln6, since there are 6 equivalent paths for

the CE to take place. The factorDS_vib accounts for the dif-ference in vibrational entropy between the saddle point and the perfect crystal configurations. From Eqs. ~5! and ~6! the self-diffusion coefficient can be written as

D52 3 a

2_ne2DG/kBT_{, ~8!}

whereDG52TDS1DH is the overall activation Gibbs free energy of the CE mechanism.

We performed constant volume MC simulations, there-fore, the TI procedure yields the Helmholtz activation free energy. First-principles calculations have shown that the ac-tivation energy of the CE mechanism is weakly affected by hydrostatic pressure.18 It is then reasonable to assume that the Helmholtz activation free energy and the Gibbs activa-tion free energy for the CE mechanism are essentially the same. In Fig. 3, we depict the calculated self-diffusion coef-ficient for three values of the temperature, which can be di-rectly compared with the experimental results ~straight lines!. The averages of the MC runs are obtained by using different samples that are spaced apart by different intervals. The error bars represent the standard deviation from the av-erage values. The error bars increase with temperature due to larger fluctuations at higher temperature. Our present results indicate that the inclusion of more degrees of freedom in the calculation through the TI formalism increases the

self-diffusion coefficient for the CE mechanism, when compared with previous estimates. These results are in reasonable agreement with the experimental data, when the scattering in the latter is taken into account. Recalling that

DF52TDS1DE, where DE is the variation in the internal

energy, we are able extract from the present results the acti-vation entropy DS for the CE mechanism. We obtained 7.860.4kB, 8.360.7kB, and 8.161.5kB at 1200, 1300, and

1500 K, respectively. Our findings for the entropy are in the experimental range of 7kB29kB.

IV. CONCLUSIONS

We have estimated the activation free energy for the CE mechanism for self-diffusion in silicon in a wide range of temperatures, using the TI method and MC simulations with an interatomic potential fitted to reproduce the most impor-tant energy paths, calculated from first principles. The TI method allows for an efficient sampling of the phase space. The self-diffusion coefficient obtained from these estimates of free energy is in reasonable agreement with experimental measurements. As expected, the inclusion of more degrees of freedom brought the calculated values for the self-diffusion coefficient closer to the experimental results than those ob-tained in previous calculations.6,8 From these results, it seems that the CE mechanism can account for a substantial part of self-diffusion in silicon.

The present results, taken together with the free energies for the defect mechanisms,2,9 suggest that the expectation that the dominant mechanism for self-diffusion could be identified from the entropic contribution, has not been ful-filled. Given the considerable error bars involved in all the calculations of the diffusion constant ~error bars of 2–3 or-ders of magnitude in the defect-mediated mechanisms9 and

FIG. 3. Self-diffusion coefficient in silicon: calculated from CE mechanism~dots! and measured by experiment ~straight lines!, see Ref. 2.

of 1–2 orders of magnitude in the present defect-free mecha-nism!, we conclude that all mechanisms may be operative, since they yield self-diffusion coefficients in a range of 1 to 2 orders of magnitude, which is comparable to the uncer-tainty in experimental measurements. In light of this conclu-sion, we expect that the mechanism by which impurities dif-fuse will depend entirely on the nature of the impurity. It seems clear now that some additional experimental probe is required to test for a possible dominant microscopic mecha-nism among those proposed for self-diffusion in silicon. One possibility is to use hydrostatic pressure and/or anisotropic stress to probe the relative importance of the various possible mechanisms of self-diffusion in Si. A study by Aziz et al.,19 using hydrostatic pressure, has shown that self-diffusion in-creases with pressure. This experiment was performed at a single temperature, and therefore, does not allow the

separa-tion of the relative changes in the activasepara-tion enthalpy and in the pre-exponential factor. Further experimental work of this type could provide valuable information on the issue of the self-diffusion mechanism in Si.

ACKNOWLEDGMENTS

One of us~A.A.! wishes to thank Harvard University for its hospitality during the course of this work, as well as to acknowledge partial support from the Brazilian funding agencies Fundo de Apoio ao Ensino e a` Pesquisa ~FAEP/ UNICAMP! and Conselho Nacional de Desenvolvimento Ci-entı´fico e Tecnolo´gico~CNPq!. This work was supported in part by Harvard’s Materials Science and Engineering Center, which is funded by NSF.

*_{Present address: Department of Physics, MIT, Cambridge, MA} 02139.

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